**Category : **6th Class

**Ratio and Proportion**

**Ratio**

Ratio of two quantities is the comparison of the given quantities. Ratio is widely used for comparison of two quantities in such a way that one quantity is how much increased or decreased by the other quantity.

For example, Peter has 20 litres of milk but John has 5 litres, the comparison of the quantities is said to be, Peter has 15 litres more milk than John, but by division of both the quantity, it is said that Peter has, \[\frac{20}{5}\text{ }=\text{ }4\]times of milk than John. It can be expressed in the ratio form as 4: 1.

Note: In the ratio\[a:\text{ }b\text{ }\left( b~\ne 0 \right)\], the quantities a and b are called the terms of the ratio and the first term (ie. a) is called antecedent and the second term (i.e., b) is called consequent.

Simplest form of a Ratio

If the common factor of antecedent and consequent in a ratio is 1 then it is called in its simplest form.

Comparison of Ratio

Comparison of the given ratios are compared by first converting them into like fractions, for example to compare 5: 6, 8: 13 and 9: 16 first convert them into the fractional form i.e.\[\frac{5}{6},\frac{8}{13},\frac{9}{16}\]

The LCM of denominators of the fractions \[=2\times 3\times 13\times 8=\text{ }624\]

Now, make denominators of every fraction to 624 by multiplying with the same number to both numerator and denominator of each fraction.

Hence,\[\frac{5}{6}\times \frac{104}{104}=\frac{520}{624},\frac{8}{13}\times \frac{48}{48}=\frac{384}{624}\] and\[\frac{9}{16}\times \frac{39}{39}=\frac{351}{624}\]. Equivalent Tractions of the given fractions are,\[\frac{520}{624},\frac{384}{624},\frac{351}{624}\]We know that the greater fraction has greater numerator, therefore the ascending order of the fractions are, \[\frac{351}{624}<\frac{384}{624}<\frac{520}{624}\] or \[\frac{9}{16}<\frac{8}{13}<\frac{5}{6}\] or 9 : 6 < 8 : 13 < 5 : 6 thus, the smallest ratio among the given ratio is 9 : 16 and greatest ratio is 5 : 6.

Equivalent Ratio

The equivalent ratio of a given ratio is obtained by multiplying or dividing the antecedent and consequent of the ratio by the same number. The equivalent ratio of \[a\,\,\times \,\,b\]is \[a\,\,\times \text{ q }:\text{ }b\text{ }\times \text{ }q\]whereas, a, b, q are natural numbers and q is greater than 1,

Hence, the equivalent ratios of 5 : 8 are,

- Example:

Mapped distance between two points on a map is 9 cm. Find the ratio of actual as well as mapped distance if 1 cm = 100 m.

(a) 10000 : 1 (b) 375 : 1

(c) 23 : 56 (d) 200 : 1

(e) None of these

Answer (a)

Explanation: Required ratio \[=\text{ }900\text{ }\times \text{ }100:\text{ }9\text{ }=\text{ }90000:\text{ }9\text{ }=-\text{ }10000:\text{ }1\]

- Example:

Consumption of milk in a day is 6 litre. Find the ratio of Consumption of milk in month of April and quantity of milk in a day?

(a) 99 : 2 (b) 30 : 1

(c) 123 : 3 (d) 47 : 3

(e) None of these

Answer (b)

Explanation: Required ratio \[=\text{ }30\text{ }\times \text{ }6:\text{ }6\text{ }=\text{ }30:\text{ }1\]

Proportion

The equality of two ratios is called proportion. If a cake is distributed among eight boys and each boy gets equal part of the cake then cake is said to be distributed in proportion. The simplest form of ratio 12: 96 is 1: 8 and 19: 152 is 1: 8 therefore,

12: 19 and 19: 152 are in proportion and written as 12 : 96 : : 19 : 152 or \[\frac{12}{96}=\frac{19}{152}\].

Terms of a Proportion

Four numbers a, b, c and d are said to be in proportion if \[\frac{a}{b}=\frac{c}{d}\] The proportion a, b, c and d is written as a : b : : c : d. In, a : b : : c : d, a and d are called extreme terms and b and c are called middle terms or means. The product of the extreme and middle terms is always equal. Therefore, \[a\text{ }\times \text{ }d\text{ }=\text{ }b\text{ }\times \text{ }c\].

Mean Proportion

If a, b and c are in continued proportion then, b is called mean proportion between a and c, and it is calculated by \[{{b}^{2}}=ac\] or b=\[\sqrt{ac}\].

- Example:

Simplify for x : 6: 18 : : 9 : x.

(a) 21 (b) 23

(c) 27 (d) 32

(e) None of these

Answer (c)

Explanation: \[6\text{ }\times \text{ }x\text{ }=\text{ }18\text{ }\times \text{ }9,\text{ }or,\text{ }x=\frac{18\times 9}{6}=\text{ }27\]

Unitary Method

The name, unitary method is self-indicating that the given variable is changed into a single unit. If the cost of more than 1 article is given and it is required to obtain the cost of x articles then the price of one article should be calculated first and the method through which the price of one article from the number of articles is obtained is called unitary method.

- Example:

The cost of 40 boxes is Rs 600. Find the cost of 20 such boxes.

(a) Rs 200 (b) Rs 300

(c) Rs 250 (d) Rs 175

(e) None of these

Answer (b)

Explanation: Cost of 40 boxes = Rs 600. Cost of 1 box \[=\text{ }Rs\,\frac{600}{40}\,~=\text{ }Rs\text{ }15\]

Cost of 20 boxes \[=\text{ }15\text{ }\times \text{ }20\text{ }=\text{ }Rs\text{ }300\]

- Example:

What will be the total cost of 25 pens and 30 pencils if the cost of 20 pencils and 45 pens are ? 160 and T 900 respectively?

(a) Rs 1060 (b) Rs 160

(c) Rs 900 (d) Rs 740

(e) None of these

Answer (d)

Explanation: Cost of 20 pencils = Rs 160. Cost of 1 pencil \[=\text{ }Rs\text{ }\frac{160}{20}=\text{ }Rs\text{ 8}\]

Cost of 45 pens = Rs 900. Cost of 1 pen \[=\text{ }Rs\text{ }\frac{900}{45}=\text{ }Rs\text{ }20\]

Cost of 25 pens and 30 pencils \[=\text{ }Rs\text{ }\left( 20\text{ }\times \text{ }25\text{ }+\text{ }8\text{ }x\text{ }30 \right)\text{ }=\text{ }Rs\text{ }\left( 500\text{ }+\text{ }240 \right)\text{ }=\text{ }Rs\text{ }740\]

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